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Converting from Other Bases to Base 10 Using the Expanded Notation
Topic Content:
- Converting from Other Bases to Base 10 Using the Expanded Notation
Worked Example 1.3.1:
Convert the following numbers to base 10 using expanded notation:
a. 107eight
b. 1211three
c. 203six
d. 4102five
e. 13011four
Solution
a. 107eight
The table shows the place values for 107. Remember that we count 0 from the last digit from right to left. We multiply each digit by the base (which is 8 for this particular question) raised to the power of the place value.
| 2 | 1 | 0 |
| 1 | 0 | 7 |
107eight = 1 × 82 + 0 × 81 + 7 × 8o
= 64 + 0 + 7
= 71ten
∴ 107eight = 71ten
b. 1211three
| 3 | 2 | 1 | 0 |
| 1 | 2 | 1 | 1 |
1211three = 1 × 33 + 2 × 32 + 1 × 31 + 1 × 3o
= 27 + 18 + 3 + 1
= 49ten
∴ 1211three = 49ten
c. 203six
| 2 | 1 | 0 |
| 2 | 0 | 3 |
203six = 2 × 62 + 0 × 61 + 3 × 6o
= 72 + 0 + 3
= 75ten
∴ 203six = 75ten
d. 4102five
| 3 | 2 | 1 | 0 |
| 4 | 1 | 0 | 2 |
4102five = 4 × 53 + 1 × 52 + 0 × 51 + 2 × 5o
= 500 + 25 + 0 + 2
= 527ten
∴ 4102five = 527ten
e. 13011four
| 4 | 3 | 2 | 1 | 0 |
| 1 | 3 | 0 | 1 | 1 |
13011four = 1 × 44 + 3 × 43 + 0 × 42 + 1 × 41 + 1 × 4o
= 256 + 192 + 0 + 4 + 1
= 453ten
∴ 13011four = 453ten
